Influence of Intermittency on the Energy Transfer Rate of Solar Wind Turbulence

The intermittency in the solar wind turbulence manifests itself in the anisotropic scaling due to the anisotropic spectral index and the intermittent level based on the extended P model. However, the influence of intermittency on the energy transfer rate remains unclear. Here we apply the partial variance of increments method to identify the intermittency for the magnetic field measurements in the fast solar wind from the Ulysses spacecraft. We distinguish the sampling direction using the angle θ RB between the local magnetic field and radial direction to study the anisotropy. We perform the multiorder structure function analyses and adopt the log-Poisson cascade model to describe the role of intermittency in the cascade process. We find that the anisotropy of the scaling becomes isotropic with a complete removal of intermittency. We compare explicitly the anisotropy of the energy transfer rate before and after removing the intermittency for the same interval for the first time. We find a distinct anisotropy with a cascade enhancement in the direction perpendicular to the local magnetic field. The removal of the intermittency greatly weakens the anisotropy by mainly reducing the perpendicular energy transfer rate. Our findings suggest that the intermittency effectively enhances the energy transfer rate, in particular in the perpendicular direction in the solar wind turbulence.


Introduction
Solar wind turbulence plays an important role in the energy transfer process in the heliosphere, as it provides a channel for the energy cascade from large scale to small scale.The turbulent fluctuations of the magnetic field in the solar wind display a probability distribution function deviating from the Gaussian distribution with a long tail, which indicates the existence of the intermittency (Bruno & Carbone 2013;Zhang et al. 2022).An effective way to identify the intermittency is the partial variance of increments (PVI) method (Greco et al. 2008), which measures the large amplitude of fluctuations.Using this method, the intermittency is found to be the site at which the solar wind is heated (Osman et al. 2011;Sioulas et al. 2022).It is of significance to study how the intermittency affects the cascade process.
Apart from the identification of intermittency using the PVI method, another useful approach to investigate the intermittency is the scaling of the multiorder structure functions S p ∼ l ξ( p) , in which l is the scale and p is the order.The scaling exponents ξ(p) have a linear relationship with the order p if there is no intermittency.Intermittency is accompanied by the multifractal scaling: the relationship is nonlinear between the scaling exponents and p.There are multiple models to explain the multifractal scaling, including the P model (Meneveau & Sreenivasan 1987), β model (Marsch & Liu 1993), lognormal model (Kolmogorov 1962), and log-Poisson model (She & Leveque 1994).These models are developed based on the solar wind observations to describe the solar wind turbulence (e.g., Ruzmaikin et al. 1995;Tu et al. 1996;Chandran et al. 2015;Mallet & Schekochihin 2017).The models based on the log-Poisson statistics successfully reproduce the results from both the numerical simulations (Müller & Biskamp 2000;Mallet et al. 2016) and the solar wind observations (Chandran et al. 2015;Zhdankin et al. 2016;Wu et al. 2023).
The log-Poisson model (She & Leveque 1994) was first proposed by writing a scaling law for the successive powers of the energy dissipation at scale l, which is where 0 < β < 1 is the characteristic of the intermittency of the turbulence.The limit β = 1 corresponds to nonintermittent turbulence, whereas the opposite limit β = 0 corresponds to an extremely intermittent state, in which the dissipation is concentrated in one single structure (Biskamp 2003).Taking the energy as δv 2 and the nonlinear interaction time being the eddy turnover time , we obtain the energy transfer rate ò l ∼ δv 3 /l.Writing the energy transfer rate at scale l as  á ñ ~t l l p p , we obtain ξ(p) = p/3 + τ p/3 .Assuming that the energy transfer time of the strongest intermittency t l ∼ l x in Equation  Dubrulle 1994;She & Waymire 1995).This log-Poisson statistics agrees well with the laboratory data of the hydrodynamic turbulence (She & Leveque 1994).Politano & Pouquet (1995) extended the log-Poisson model to the MHD turbulence and found a good agreement with the solar wind far from the Sun (Burlaga 1991).The extended log-Poisson model takes into account that the energy transfer time to small scales is now measures the efficiency of the nonlinear interactions, in which 21 2 is the eddy turnover time (δB is the magnetic fluctuations in Alfvén units), τ A = l/B 0 is the Alfvén time (B 0 is the background magnetic field in Alfvén units), and a is an arbitrary power law.Therefore, taking the energy as (δB 2 + δv 2 ), the energy transfer rate is In particular, in the K41 theory (Kolmogorov 1941), a = 0 and g = 3, while in the IK theory (Iroshnikov 1964;Kraichnan 1965), a = 1 and g = 4. Now we get The energy transfer rate does not change with scale yielding Müller et al. (2003) detached the nonlinear transfer time with t l , fixed t l ∼ l x to the K41 timescale, that is, t l ∼ l/δB ∼ l (1−1/ g) , then x = 1 − 1/g and obtained the following relationship: Recently Wu et al. (2023) found that Equation (6) agrees well with the the observational scaling exponents in the near-Sun solar wind turbulence with C = 1, g = 4.
In order to further illustrate the consequence of intermittency, an effective way is to observe the scaling after removing the intermittency.The Kolmogorov scaling was recovered from the removal of intermittency for the magnetic field both in the solar wind at 1 au (Salem et al. 2009) and at 0.17 au (Wu et al. 2023).Wang et al. (2014) found that the spectral index becomes isotropic without the intermittency.Pei et al. (2016) further investigated scaling of the magnetic-trace structure functions and utilized the extended P model developed by Tu et al. (1996), whose parameters are spectral index and the occupied fraction P, to describe the effect of intermittency on the scaling behavior.They found that the multifractal scaling becomes monoscaling and the anisotropy of the spectral index is evidently weakened with a not-complete removal of intermittency.However, the influence of intermittency on the energy transfer rate has not been discussed yet.
Here, we investigate the influence of intermittency on the energy transfer rate in the solar wind turbulence by evaluating the energy transfer rate under the framework of the extended log-Poisson cascade model.We calculate the multiorder structure functions and analyze the scaling before and after the removal of intermittency identified by the PVI method.We find that the energy transfer rate enhances in the direction perpendicular to the local magnetic field and when the intermittency is removed, the energy transfer rate and its anisotropy are both weakened.In Section 2, we describe the Ulysses measurements and the method to calculate original and conditioned structure functions and energy transfer rates.In Section 3, we show both the original and conditioned scaling exponents and their corresponding energy transfer rates.In Section 4, we discuss our results and draw our conclusions.

Data and Method
The Ulysses spacecraft measures the magnetic field by Vector Helium Magnetometer (Balogh et al. 1992) with a time resolution of 1 s.We utilize the measurements in a fast solar wind stream from 1995 May 1 to 1995 May 10 with the average flow speed for the 10 days V 0 = 771 km s −1 obtained at r ∼ 1.48 au and latitude f ∼ 47°.This interval was observed by the Ulysses spacecraft during a polar pass.It is time stationary and does not contain the corotating interaction regions, sector boundary crossings of the heliospheric current sheet, interplanetary coronal mass ejections, etc.Therefore, the presented results are not affected by the large magnetic structures.
The vector magnetic field increment is calculated by , 7 1 2 where τ is the time lag, B 1 = B(t) and B 2 = B(t + τ).The magnetic-trace structure functions with pth-order S p (τ) are defined as the pth-order moments of the magnetic field increments p p in which 〈 〉 denotes an ensemble time average of the entire 10 days.The sampling angle θ RB (t, τ) is obtained between the radial direction and the local background magnetic field B l = (B 1 + B 2 )/2 for each pair of B 1 and B 2 .Any angles greater than 90°are reflected below 90°.In order to analyze the anisotropy, we arrange the increments into nine groups according to their corresponding θ RB with 10i°< θ RB < 10i°+ 10°, i = 0, 1, 2,K,8 and obtain their original structure functions S p (τ) respectively.
We apply the PVI method to measure the intermittency, which can be classified by setting a threshold on the PVI values.The PVI is defined in terms of the magnetic field increment as . 9 2 Another parameter to evaluate the intermittency is the flatness F, Flatness of a Gaussian-distributed variable is equal to 3, and intermittency refers to the raised tails of the distribution that deviate from that of a Gaussian distribution.We adopt the same recursive method to determine the required threshold as was done by Wu et al. (2023) to identify all the intermittency for each group, and therefore, we can perform a clean removal of intermittency.We gradually lower PVI thres and remove these increments with > PVI PVI thres until F calculated using remaining increments is equal to 3. The fraction is obtained as the ratio between the number of increments with > PVI PVI thres and the total number of increments.It is less than 6% for each group.We further obtain the conditioned structure functions ( ) t S c p using the leftover increments for nine groups.
We perform linear fits to S p (τ) and ( ) t S c p in the log-log space within 10 < τ < 100 s, except the S p (τ) with 0°< θ RB < 10°(for which the range is smaller).The range is determined in which the structure functions show the powerlaw behavior.The linear slopes are the estimation of the scaling exponents ξ(p).We set C = 1 and C = 2 in Equation (6), and then fit ξ(p) using Equation (6) to obtain the value of g.
We utilize Equation (3) to evaluate the energy transfer rate.Since the plasma data has a low resolution of 4 minutes, we cannot obtain δv within the range 10 < τ < 100 s.The Alfvén ratio γ A = δv 2 /δB 2 = 0.65 for this fast solar wind interval.Therefore, we approximate ( ) 2 21 2 to be 1.28 δB.The energy transfer rate can be calculated as Note that ò evaluates the relative value of the energy transfer rate.Here we use the unit nT for the magnetic field and use the timescale as the spatial scale under the Taylor hypothesis in the calculation process.We calculate ò and ò c using the corresponding δB and g obtained from all increments and the increments with < PVI PVI thres , respectively, for the nine groups observed in different directions.

Results
Figure 1 shows the original (left) and conditioned (right) magnetic-trace structure functions of two groups measured in the parallel (0°< θ RB < 10°, top) and perpendicular (80°< θ RB < 90°, bottom) direction.Both S p (τ) and ( ) t S c p present the power-law shape within the inertial range marked by blue shadows.The procedure to remove the intermittency affects the scaling of the structure functions for both groups.The conditioned structure functions of two groups become similar, while the original structure functions show remarkable differences.
The scaling exponents obtained from the original and conditioned structure functions are displayed by the viridis and red dots for nine groups with different sampling angles in Figure 2. The range within which we perform the linear fits shows good power-law behavior for all structure functions.Therefore, it is reasonable to obtain the scaling exponents within this range.The original scaling shows an obvious monotonically growing concave shape, reflecting the influence of the intermittency on the cascade process.From 0°< θ RB < 10°to 80°< θ RB < 90°, ξ B (5) of the conditioned structure functions decreases monotonically.The structure functions become flatter gradually as the sampling angle becomes larger.While the intermittency is cleanly removed, a monoscaling is recovered and the linear coefficients are very close for different directions.The second-order scaling exponents correspond to the spectral index α with α = − ξ (2) − 1.The spectral index anisotropy is obvious and disappears when the intermittency is removed, which are consistent with previous results (Wang et al. 2014;Pei et al. 2016).
We compare the model predictions with the original scaling exponents for both C = 1 and C = 2.It is clear that the predictions with C = 1 (green dashed lines) are closer to the observational results compared to that of C = 2 (blue dashed lines).The model with C = 1 well reproduces the nonlinear behavior of ξ(p) shown in Figure 2, which is consistent with the result in Wu et al. (2023) and implies that the intermittency is probably 2D current sheet-like structures.We show g estimated from the observational scaling through the curve fitting with C = 1 for Equation (6) in the right panel of Figure 2.
The original g shows an obvious anisotropy.From 0°< θ RB < 10°to 80°< θ RB < 90°, g gradually grows from 2.48 to 4.83, jumping across 3 with θ RB around 20°and across 4 with θ RB around 40°.This indicates that the energy transfer time t t = R tr nl eff is smaller (a < 0, R eff < 1) than the eddy turnover time in the parallel direction and larger (a > 0, R eff > 1) than the eddy turnover time in the perpendicular direction.Since the scaling is linear after the removal of intermittency, we set τ p = 0 , obtain the linear slopes by leastsquares regression, and show them in red dots in the right panel of Figure 2. The conditioned g illustrates the isotropy and the value of g is around 2.44, which is less than 3 and close to the original g in parallel direction.This indicates that the energy transfer time is smaller (a < 0, R eff < 1) than the eddy turnover time in all directions when the intermittency is absent.
To demonstrate the influence of the intermittency on the energy transfer rate, we present the relative energy transfer rate versus timescale τ measured in different directions in Figure 3.The energy transfer rates of both the original and conditioned fluctuations are nearly constant at different τ in all directions, which is consistent with the hypothesis that the energy transfer rate does not change with the scale.We average the energy transfer rate within 10-100 s and show the averaged energy transfer rate in Figure 4.It is obvious that the original energy transfer rate is larger in the perpendicular direction than in the parallel direction, with the ratio between them around 15.The removal of intermittency greatly weakens the energy transfer rate in all directions.It reduces the energy transfer rate to 50.0% in the parallel direction and to 9.7% in the perpendicular direction.The resulting ratio between the perpendicular and the parallel directions decreases to around 3. This reflects that the intermittency enhances the energy transfer rate, especially in the perpendicular direction, leading to a greater anisotropy.

Conclusions and Discussion
In this work, we investigate the influence of intermittency on the energy transfer rate using the extended log-Poisson cascade model based on Ulysses observations.We classify intermittency by the recursive PVI method and find the maximum fraction is around 5%.We compare the multiorder scaling of the magnetic-trace structure functions before and after the removal of intermittency.The original scaling can be well reproduced by the log-Poisson cascade model assuming the intermittent structures is 2D sheet-like.The conditioned scaling shows good isotropy.We evaluate the energy transfer rate using the information from the observed scaling and find the distinct anisotropy.The ratio of the energy transfer rate between the perpendicular and parallel direction reaches 15.After removing the intermittency, the energy transfer rate reduces to 50.0% in the parallel direction and to 9.7% in the perpendicular direction.The removal of intermittency greatly weakens the anisotropy of the energy transfer rate.
The interval analyzed in this work is in the fast solar wind stream.The fluctuations have some distinct features between the fast and solar wind wind.For example, the 2D selfcorrelation function level contours display different anisotropy at 1 au (Dasso et al. 2005); the slow streams do not show a clear low-frequency break on the power spectrum (Bruno et al. 2019) as the fast streams; the sign of the energy cascade rate given by third-order scaling law strongly relates to the solar wind speed (Wu et al. 2022).It is possible that the turbulence in the slow solar wind is not influenced in the same way by the intermittency as we show in the work, which is worth further investigation in future.
The energy transfer process is fundamental to understanding the solar wind turbulence.Multiple approaches are developed to evaluate the exact energy transfer rate (Wu et al. 2023).The studies on the anisotropy of the energy cascade rate suggest that the energy transfer rate is larger in the perpendicular direction.MacBride et al. (2008) found that the energy cascade rate is smaller in the parallel cascade than in the perpendicular cascade at 1 au using the third-order moments methods, assuming the energy transfer occurs only along these two directions.Adhikari et al. (2022) obtained the same conclusion for turbulent cascade rates based on the formula derived through dimensional analysis.This anisotropy could be interpreted as a major 2D component in the solar wind turbulence described by the nearly incompressible MHD (Zank et al. 2017).In this Letter, we delve into the influence of intermittency on the cascade rate for the first time and present the anisotropy both before and after the removal of intermittency.More investigation related to the nature of the anisotropic energy transfer rate is required.
The radial variations of the magnetic field intermittency are investigated from many perspectives.Chhiber et al. (2021) showed multifractal scaling in the inertial range and at the subproton scale in the near-Sun solar wind.Cuesta et al. (2022) presented the radial evolution of intermittency by examining the behavior of kurtosis from 0.16 to 10 au combining the observations of the Parker Solar Probe, Helios 1, and Voyager 1. Sioulas et al. (2022)  p with the intermittency removed.The blue, orange, green, red, and purple represent the S p with order p = 1, 2, 3, 4, 5, respectively.The blue shadow denotes the range within which the scaling exponents are obtained by fitting.Bottom: magnetic-trace structure functions vs. timescale τ measured in the direction perpendicular to the local magnetic field, with the same formats as the top panels.observed an increasing efficiency when moving away from the Sun using the singularity width as a measure for the efficiency of the nonlinear energy cascade mechanism.These previous works suggest that the magnetic field intermittency in the solar wind is observed to radially strengthen.
When analyzing the anisotropy, it is important to track the changes in sampling direction in the measurements of spacecraft (Cuesta et al. 2022) and the radial evolution could be a result of measurement effect that more comparatively highly intermittent perpendicular intervals are sampled by the probes with increasing distance, an effect related directly to the  evolution of the Parker spiral (Sioulas et al. 2022;Zank et al. 2022).Adhikari et al. (2022) confirmed this effect on the anisotropy and found that solar wind fluctuations are well described by the nearly incompressible MHD.Adhikari et al. (2022) calculated the anisotropic turbulent cascade rates and found that the perpendicular component has the largest rate.Our results suggest that intermittency plays a role in the enhancement of the perpendicular energy transfer rate.Our findings are consistent with these results and underline the importance of intermittency in the energy transfer process of the solar wind turbulence.How the intermittency contributes to the overall radial variations of the energy transfer rates requires further study.
(1) leads to τ p = − xp + C − Cβ p and C = x/(1 − β) (C represents the codimension of the intermittent structures; Figure 1.Top: magnetic-trace structure functions vs. timescale τ measured in the direction parallel to the local magnetic field.(a) Original S p ; (b) conditioned S cp with the intermittency removed.The blue, orange, green, red, and purple represent the S p with order p = 1, 2, 3, 4, 5, respectively.The blue shadow denotes the range within which the scaling exponents are obtained by fitting.Bottom: magnetic-trace structure functions vs. timescale τ measured in the direction perpendicular to the local magnetic field, with the same formats as the top panels.

Figure 2 .
Figure 2. Left: scaling exponents ξ of the original (colors in viridis map denote θ RB ) and conditioned (red) structure functions for the magnetic field as a function of the order p measured in different ranges of θ RB .The green (blue) dashed lines denote the predictions of the log-Poisson intermittency model with C = 1 (C = 2) but different g.Right: dependence of g on θ RB for the original (viridis colors) and conditioned (red) scaling exponents.The error bar shows the standard errors.The blue dashed line corresponds to g = 2.44.The black dashed line corresponds to g = 3 and g = 4.

Figure 3 .
Figure 3. Energy transfer rate of the original (left) and conditioned (right) fluctuations for the magnetic field as a function of τ measured in different ranges of θ RB with the color shown in the color bar.